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A002652
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Theta series of Kleinian lattice Z[ (-1+sqrt(-7))/2 ] in 1 complex (or 2 real) dimensions.
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8
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1, 2, 4, 0, 6, 0, 0, 2, 8, 2, 0, 4, 0, 0, 4, 0, 10, 0, 4, 0, 0, 0, 8, 4, 0, 2, 0, 0, 6, 4, 0, 0, 12, 0, 0, 0, 6, 4, 0, 0, 0, 0, 0, 4, 12, 0, 8, 0, 0, 2, 4, 0, 0, 4, 0, 0, 8, 0, 8, 0, 0, 0, 0, 2, 14, 0, 0, 4, 0, 0, 0, 4, 8, 0, 8, 0, 0, 4, 0, 4, 0, 2, 0, 0, 0, 0, 8, 0, 16, 0, 0, 0, 12, 0, 0, 0, 0, 0, 4, 4, 6, 0
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| In other words, theta series of lattice with Gram matrix [2, 1; 1, 4].
The number of integer solutions (x,y) to x^2+xy+2y^2=n.
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REFERENCES
| J. H. Conway and N. J. A. Sloane, Complex and integral laminated lattices, Trans. Amer. Math. Soc., 280 (1983), 463-490.
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LINKS
| John Cannon, Table of n, a(n) for n = 0..5000
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FORMULA
| G.f.: theta_3(z)*theta_3(7z) + theta_2(z)*theta_2(7z).
G.f. A(x) satisfies 0=f(A(x), A(x^2), A(x^4)) where f(u, v, w)=u^2+5v^2+4w^2+2uw-4uv-8vw. - Michael Somos, Sep 20 2004
G.f.: ((eta(q)eta(q^7))^3+4(eta(q^2)eta(q^14))^3)/(eta(q)eta(q^2)eta(q^7)eta(q^14)). - Michael Somos May 28 2005
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6) = u1^3*u6 + 2*u2^3*u3 + 18*u1*u3*u6^2 + 18*u2*u3^2*u6 + 6*u1*u2^2*u6 + 3*u1^2*u2*u3 - 3*u2*u3^3 - 18*u2*u3*u6^2 - 6*u1*u6^3 - 9*u1*u3^2*u6 - 6*u1*u2^2*u3 - 6*u1^2*u2*u6 - Michael Somos Jun 03 2005
Moebius transform is period 7 sequence [2, 2, -2, 2, -2, -2, 0, ...]. - Michael Somos Oct 07 2005
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EXAMPLE
| 1 + 2*q^2 + 4*q^4 + 6*q^8 + 2*q^14 + 8*q^16 + 2*q^18 + 4*q^22 + 4*q^28 + 10*q^32 + 4*q^36 + 8*q^44 + 4*q^46 + 2*q^50 + 6*q^56 + 4*q^58 + 12*q^64 + 6*q^72 + ...
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MATHEMATICA
| f[d_] := KroneckerSymbol[-7, d]; a[n_] := 2*Total[f /@ Divisors[n]]; a[0]=1; Table[a[n], {n, 0, 101}] (* From Jean-François Alcover, Nov 08 2011, after Michael Somos *)
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PROG
| (PARI) a(n)=local(t2, t3); if(n<1, n==0, t2=2*sum(n=1, (sqrtint(max(0, 4*n-7))+1)\2, x^(n*n-n)); t3=1+2*sum(n=1, sqrtint(n), x^(n*n)); polcoeff(t3*subst(t3, x, x^7)+x^2*t2*subst(t2, x, x^7), n))
(PARI) a(n)=local(t); if(n<1, n==0, 2*issquare(n) +2*sum(y=1, sqrtint(n*4\7), 2*issquare(t=4*n-7*y^2)-(t==0))) /* Michael Somos, Sep 20 2004 */
(PARI) {a(n)=local(A, A1, A2); if(n<0, 0, A=x*O(x^n); A1=eta(x+A)*eta(x^7+A); A2=eta(x^2+A)*eta(x^14+A); polcoeff( (A1^3+4*x*A2^3)/(A1*A2), n))} /* Michael Somos May 28 2005 */
(PARI) a(n)=if(n<1, n==0, 2*qfrep([2, 1; 1, 4], n, 1)[n]) /* Michael Somos Jun 03 2005 */
(PARI) a(n)=if(n<1, n==0, 2*sumdiv(n, d, kronecker(-7, d))) /* Michael Somos Oct 07 2005 */
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CROSSREFS
| a(n)=2*A035182(n) if n>0.
Sequence in context: A197513 A097666 A144810 * A202541 A070676 A068451
Adjacent sequences: A002649 A002650 A002651 * A002653 A002654 A002655
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KEYWORD
| nonn,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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